Segments Formed by a Tangent and a Secant

Segments Formed by a Tangent and a Secant

How to solve the segments formed by a tangent and a secant problems: formula, proof, example, and its solution.

Formula

If a tangent and a secant start from the same point, then the square of the tangent segment is equal to the product of the secant's segments.

(purple)2 = (blue)⋅(dark blue)

(purple), (dark purple): segment from the tangent
(blue), (dark blue): segments from the secant

Proof

Segments Formed by a Tangent and a Secant: Proof of the Formula

Draw the green arc
and two additional white chords like above.

The upper red angle is the inscribed angle
of the green intercepted arc.

The lower red angle
is the angle formed by a tangent and a chord,
whose intercepted arc is the same green arc.

Both red angles have the same intercepted arc.
m∠(red) = (1/2)⋅m[green arc]

So the red angles are congruent.

Draw the red dot angle
at the intersecting point of the tangent and the secant.

See these two triangles.
These two have two pairs of congruent angles.

So, by the AA similarity,
these two triangles are similar triangles.

So the corresponding sides are proportional:
(ratio of the longest sides) = (ratio of the shortest sides).

Similarity of sides in triangles

Then (purple)2 = (blue)⋅(dark blue).

Proportion

Example

Find the value of x. Tangent segment: x. Secant segments: 4, 5.

[x] and [4, (4 + 5)]

x2 = 4(4 + 5)